From Localized Packets to Plane Waves: A Time-Domain Approach to Transport in Mesoscopic Systems

Abstract: Quantum transport in mesoscopic systems is conventionally formulated within the Landauer--Büttiker scattering framework, where steady-state currents emerge from the transmission of plane waves representing propagating carriers. While highly successful, this description obscures the explicit time-domain dynamics of individual fermionic excitations and their role in establishing macroscopic transport. Here, we present an exact and self-contained time-domain construction of Landauer transport based on a discrete basis of orthogonal fermionic wave packets. Starting from a second-quantized formulation, we define packet creation operators via a continuous Fourier transform over a finite transport energy window. By encoding the Pauli exclusion principle, which enforces a fundamental temporal spacing $Δt = h/eV$, the current is reproduced in terms of orthogonal wave packets that are used for the rigorous construction of the many-body fermionic state. In this representation, a noiseless current emerges as a deterministic sequence of charge-carrying events, yielding the Landauer conductance $G_0 = e^2/h$ without invoking momentum-space kinematics. We further demonstrate that this construction remains exact for arbitrary energy dispersion. Additionally, the underlying Fock space decomposition into finite disjoint energy sub-bands renders the numerical approach highly scalable for high performance computing platforms. Our results establish a direct and rigorous bridge between the continuous scattering description of quantum transport and a discrete, time-resolved picture based on fermionic wave packets.

• The paper introduces a novel time-domain formalism using wave packets to derive conductance quantization and non-equilibrium steady states.
• The methodology employs an orthogonal basis of Wannier-type states, ensuring accurate, scalable simulation of mesoscopic transport under dynamic and non-linear conditions.
• Numerical results confirm deterministic packet emission and validate the approach against established Landauer-Büttiker predictions, highlighting its computational efficiency.

Time-Domain Wave-Packet Formalism for Quantum Transport in Mesoscopic Systems

Overview

The paper "From Localized Packets to Plane Waves: A Time-Domain Approach to Transport in Mesoscopic Systems" (2606.19012) establishes a rigorous framework for quantum transport in mesoscopic conductors based on a time-domain representation of fermionic wave packets. This approach stands in contrast to the prevailing Landauer-Büttiker (L-B) formalism, which is fundamentally energy-domain and obscures explicit temporal dynamics. By constructing an orthogonal basis of time-localized fermionic states, the work directly derives conductance quantization and non-equilibrium steady states (NESS), maintains unconditional validity for arbitrary dispersion relations, and demonstrates strong practical advantages including efficient parallelizability for high-performance numerical studies.

Theoretical Construction

The key theoretical innovation is the formulation of transport through a second-quantized basis of orthogonal Wannier-type wave packets, each defined by a packet creation operator via the continuous Fourier transform over a finite transport window. The Pauli exclusion principle enforces a minimum temporal separation $\Delta t = h/eV$ between subsequent packet emissions, imposing a granular time lattice without invoking any explicit momentum-space structure. The resulting basis is strictly orthogonal and completely spans the accessible transport space, thereby providing a one-to-one correspondence between the discrete sequence of charge-carrying events (packets) and macroscopic current.

Unlike traditional approaches that rely on steady-state plane waves, this time-domain description offers:

• Exact mapping between the continuous scattering (L-B) formalism and a discrete, time-resolved packet basis.
• Non-perturbative derivation of conductance quantization ($G_0 = e^2/h$), with the quantized current realized as a strictly deterministic sequence of orthogonal, ballistic packet emissions.
• Full validity for arbitrary dispersion $E(k)$, rigorously preserving orthonormality and current quantization independently of underlying band structure.

Numerical Implementation and Performance

The general framework is operationalized through efficient numerical algorithms. The time-dependent Schrödinger equation is solved using the Crank-Nicolson scheme within a finite simulation domain augmented by complex absorbing boundary conditions. The transport window is partitioned into energy sub-bands; packets from each can be propagated independently, enabling strict $O(N_t)$ scaling and straightforward massive parallelization. This is in direct contrast to memory-intensive approaches such as NEGF, which require global integration over the simulation history.

At finite temperatures, the energy window decomposition becomes an approximation controlled by Fermi-Dirac statistics; each sub-band defines a statistically weighted, independent packet train. The approach naturally accommodates arbitrary time-dependent driving scenarios (e.g., bias modulation, AC fields), as time-dependent occupation weights can be assigned on a per-packet basis.

Benchmarking and Results

Formation of DC Currents and NESS

Numerical reconstructions confirm that the injection of a sufficiently dense packet train results in a noiseless DC current; fluctuations and errors in the reconstructed current scale as $1/N$ with the number of packets. Both linear (constant group velocity) and parabolic (energy-dependent group velocity) regimes are analyzed, with rigorous maintenance of current quantization and orthogonality in both.

Resonant Scattering and Quasi-Bound States

The packet formalism enables direct, time-domain observation of NESS plateau formation and decays in systems with resonant tunneling structures. Tunneling lifetimes, resonance energies, and transmission coefficients are extracted via wave-packet dynamics and show excellent quantitative agreement with conventional stationary approaches (QTBM/Kwant).

Application to Resonant Tunneling Diodes

For non-linear transport in resonant tunneling diodes (RTDs), the formalism efficiently simulates both static and high-frequency dynamic responses, including drive frequencies in the 100 GHz regime. The methodology reproduces standard effects such as negative differential resistance. Crucially, it enables extraction of dynamic admittance—conductance and susceptance—directly from driven time-domain currents, resolving the capacitive-to-inductive transitions associated with quantum dwell times.

Implications and Theoretical Significance

Unification of Temporal and Energy-Domain Perspectives

By bridging the gap between the temporal perspective of electron quantum optics and the energy-domain L-B formalism, the approach clarifies the microscopic origin of conductance quantization and the role of the Pauli exclusion principle in enforcing noiseless ballistic transport.

Practical Utility

The method's favorable computational scaling and intrinsic parallelizability position it as a powerful alternative for scenarios where explicit time-resolved dynamics, NESS formation, or high-frequency transport properties are of interest. Its strict mapping between energy and time domains makes it well-suited for dynamic device modeling and experimental setups involving time-dependent signals or pulsed excitations.

Limitations and Prospects

• The framework is presently optimized for 1D or quasi-1D ballistic systems with negligible interactions. Extension to multi-dimensional or strongly correlated systems is computationally prohibitive with current memory and processor architectures.
• Mean-field interactions (e.g., via time-dependent Poisson coupling) are not naturally incorporated. For these, established alternatives such as NEGF may remain preferable.
• The approach may be further extended by leveraging emerging high-performance computing platforms, potentially enabling 2D/3D generalizations or inclusion of interacting effects in future work.

Conclusion

The time-domain, wave-packet-based formalism introduced here (2606.19012) constitutes a rigorous, computationally efficient method for modeling quantum transport in mesoscopic systems. It provides a direct, transparent mapping between microscopic time-resolved particle injection and macroscopic stationary transport observables. The approach reproduces all key features of the Landauer-Büttiker formalism, extends naturally to dynamic and thermally driven scenarios, and is benchmarked with high fidelity against conventional methods. This theoretical and computational advance is significant for ongoing efforts in time-resolved quantum transport and the modeling of dynamic mesoscopic devices, with promising avenues for future extension as computational resources expand.